A226519 Irregular triangle read by rows: T(n,k) = Sum_{i=0..k} Legendre(i,prime(n)).

1, 1, 0, 1, 0, -1, 0, 1, 2, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, -1, -2, -1, 0, -1, 0, 1, 2, 1, 2, 1, 0, -1, 0, 1, 0, -1, -2, -1, -2, -1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, -1, 0, 1, 0, 1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 1, 0

Offset: 1

Author

N. J. A. Sloane, Jun 19 2013

Keywords

• sign
• tabf

Comments

Strictly speaking, the symbol in the definition is the Legendre-Jacobi-Kronecker symbol, since the Legendre symbol is defined only for odd primes.

Examples

			Triangle begins:
  1;
  1, 0;
  1, 0, -1, 0;
  1, 2,  1, 2, 1, 0;
  1, 0,  1, 2, 3, 2,  1,  0,  1, 0;
  1, 0,  1, 2, 1, 0, -1, -2, -1, 0, -1,  0;
  1, 2,  1, 2, 1, 0, -1,  0,  1, 0, -1, -2, -1, -2, -1, 0;
  ...
		

References

• József Beck, Inevitable randomness in discrete mathematics, University Lecture Series, 49. American Mathematical Society, Providence, RI, 2009. xii+250 pp. ISBN: 978-0-8218-4756-5; MR2543141 (2010m:60026). See page 23.

Links

• G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Crossrefs

A variant of A226518, which is the main entry for this triangle.

Cf. A165582, A226518.

Programs

Magma

A226519:= func< n,k | n eq 1 select k else  (&+[JacobiSymbol(j, NthPrime(n)): j in [0..k]]) >;
[A226519(n,k) : k in [1..NthPrime(n)-1], n in [1..15]]; // G. C. Greubel, Oct 05 2024

Maple

with(numtheory);
T:=(n,k)->add(legendre(i,ithprime(n)),i=1..k);
f:=n->[seq(T(n,k),k=1..ithprime(n)-1)];
[seq(f(n),n=1..15)];

Mathematica

Table[P = Prime[n]; Table[JacobiSymbol[k,P], {k,P-1}]//Accumulate, {n,15}]// Flatten (* G. C. Greubel, Oct 05 2024 *)

SageMath

def A226519(n,k): return k if n==1 else sum(jacobi_symbol(j, nth_prime(n)) for j in range(k+1))
flatten([[A226519(n,k) for k in range(1,nth_prime(n))] for n in range(1,16)]) # G. C. Greubel, Oct 05 2024